Koszul Duality Patterns in Representation Theory

摘要: The aim of this paper is to work out a concrete example as well provide the general pattern applications Koszul duality repre- sentation theory. consists three parts relatively independent each other. first part gives reasonably selfcontained introduction rings and duality. are certain Z-graded with particularly nice homological properties which involve kind Thus, ring one associates naturally dual ring. second devoted an application representation theory semisimple Lie algebras. We show License or copyright restrictions may apply redistribution; see http://www.ams.org/journal-terms-of-use KOSZUL DUALITY PATTERNS 527 that block Bernstein-Gelfand-Gelfand category O corresponds any fixed central character governed by Moreover, governs subcategory again. This generalizes selfduality theorem conjectured Beilinson Ginsburg in 1986 proved Soergel 1990. In third we study cate- gories mixed perverse sheaves on variety stratified affine linear spaces. criterion for such be flag case reduces setup two. more manifolds Grassmannians should yield interesting results about representations quantum groups Department Mathematics, Massachusetts Institute Technology, Cambridge, 02139 E-mail address: sasha@math.mit.edu University Chicago, Illinois 60637 ginzburg@math.uchicago.edu Max-Planck-Institut fur Mathematik, Gottfried-Claren-Strase 26, D-53 Bonn 3, Germany Current Mathematisches Institut, Universitat Freiburg, Albertstrase 23b, D-79104 soergel@sun1.mathematik.uni-freiburg.de